Short answer: yes
Lets talk about what all those grounds are for. It is not so the electrons can go underground and party.

In a correctly wired modern home you have two ungrounded conductors (hot wires) and one grounded conductor (Neutral). The neutral wire is derived by attaching it at the midpoint of the windings on the transformer for the two hots. The neutral is also attached to the ground by the power company(POCO) before it gets to your house. In side the house the electrician runs his (or her

) branch circuits. In our example circuit for a family room, he will run a hot, a neutral, and a ground. Where the POCO's wires enter the house the ground wire and the neutral wire come together with the neutral wire from the POCO.
When a fault occurs, say in the ceiling fan of this living room (and by fault, I mean a hot wire touches a ground wire or something it is attached to) the ground wire provides a low impedance path back to the SOURCE. What is this SOURCE Andy speaks of? Put simply it is the midpoint of the windings on the POCO transformer mentioned above. The electrons want to go back to the source to party. It has not one honker thing to do with the plant Earth or those groundrods you have in the backyard.
So why is it so critical to have all these grounds connected together??? If the fault in the ceiling fan were to travel back to a receptacle and the ground that was attached to the SOURCE was not attached to the ground in the ceiling fan then all the metal parts on that recp. would now be hot. And say you were barefoot and standing on a grounded heating grate and you reached down and touched the screw holding the plate on you would likely be killed.
Now let's prove some of this with OHM's law.
In the NEC the electrician must drive a ground rod. If the ground rod has greater than 25 ohms resistance, he must drive another one. So in the NEC 25 ohms is the magic number.
I=E/R Where I=Amps E=Volts R=resistance
If we were to depend on the ground rod without it being attached to the SOURCE lets see what would happen on the ceiling fan example...
I= 120V/25ohms = 4.8Amps You would have 4.8A potential on that ground screw. On a 15Amp breaker 4.8A ain't gonna make the breaker trip. It takes MILLIAMPS to kill you.
If we were to attach the ground wires from the ceiling fan back to the source then lets see what would happen
I=E/R 120V/.001ohms(the resistance of the wire which by the way I'm pulling this number out of my butt because I don't know the resistance of the wire, but it is accurate enough to prove a point.) = 120,000A Which is enough to trip the 15A breaker instantly saving you and your family.
Did anyone learn anything, or was that a waste of time???